On coherent situations
Sjoerd Crans – 19 February 1997
Mike Johnson has been claiming recently that he understands the coherence conditions for associativities and identities. In order to verify that claim, I went through Chapter 4 of his PhD thesis (which is as yet unavailable).
(Internal) coherence is about commutativity of formal diagrams of natural transformations, Mike uses pasting schemes to describe coherent situations. Example: coherence for multiplication m which is associative up to an associator a, pentagon. A situation consists of a collection of k-dimensional pasting schemes (thought of as ways of multiplying n things) and a collection of singleton (k+1)-dimensional pasting schemes (the things for which one wants to establish coherence), together with a realization, satisfying conditions. The situation is coherent if all (k+1)-dimensional composites with same domain and codomain are equal ("all diagrams of a's commute"). To make the example into a situation, take 2-dimensional simplicial sets and singleton 3-dimensional simplicial sets. One needs a realization of these in an \omega-category, which is taken to be: one object, arrows natural numbers, 2-cells functors Kn -> Km, 3-cells natural transformations, with realization: triangles sent to m, tetrahedra to a. To prove that all diagrams commute from commutativity of the pentagon, and more generally, to get sufficient conditions for coherence, Mike uses commuting completions of forks, and the well-known no overlap argument.
How useful is this theory of coherence in relation to weak n-categories?
(Weakly) monoidal n-categories goes via (n+1)-dimensional simplicial sets, as Mike shows. For Gray-categories with a multiplication K tensor K -> K, the canonical things are pseudo-natural transformations, and the thing to realize in is not an n-category, but some Gray-version of these. If the pasting theorem can be extended to these sort of higher-dimensional categories, then this theory of coherence extends to this situation too. For bicategories can just take the \omega-category to have more objects and arrows strings of objects, etc., and coherence of the associativity of composition can be dealt with as before. For tricategories, 0-composition is a homomorphism of bicategories, which is too weak for the present set-up. So need coherence locally and of homomorphisms before can do coherence of 0-composition. This inductive approach is the same as in GPS's "Coherence for tricategories". Making the identities of the same shape as the multiplication only works in a higher-dimensional category which itself has strict identities.
Conclusion: Mike's approach to associativities and identities has the potential to go quite a long way. What it certainly can not do is relate associativities and identities in more than one dimension at once.